On a girth-free variant of the Bourgain-Gamburd machine

Approximate groups and their applications: work of Bourgain, Gamburd, Helfgott and Sarnak

This is a survey of several exciting recent results in which techniques originating in the area known as additive combinatorics have been applied to give results in other areas, such as group theory, number theory and theoretical computer science. We begin with a discussion of the notion of an approximate group and also that of an approximate field, describing key results of Freiman-Ruzsa, Bourgain-Katz-Tao, Helfgott and others in which the structure of such objects is elucidated. We then move on to the applications. In particular we will look at the work of Bourgain and Gamburd on expansion p…

Prime Points in Orbits: Some Instances of the Bourgain-Gamburd-Sarnak Conjecture

We use Vaughan's variation on Vinogradov's three-primes theorem to prove Zariski-density of prime points in several infinite families of hypersurfaces, including level sets of some quadratic forms, the Permanent polynomial, and the defining polynomials of some pre-homogeneous vector spaces. Three of these families are instances of a conjecture by Bourgain, Gamburd and Sarnak regarding prime points in orbits of simple algebraic groups. Our approach is based on the formulation of a general condition on the defining polynomial of a hypersurface, which suffices to guarantee that Zariski-density of…

Markoff 三元组模 $p$ 提升的界限

Bounding Lifts of Markoff Triples mod $p$

本文通过分析 Markoff 模 $p$ 图中路径的增长，为 Markoff 曲面模 $p$ 点的提升提供了两种上界估计，并给出了改进这些界限的数值证据与启发式论证。

Explicit growth and expansion for SL_2

We give explicit versions of Helfgott's Growth Theorem for $\SL_2$, as well as of the Bourgain-Gamburd argument for expansion of Cayley graphs modulo primes of subgroups of $\SL_2(\Zz)$ which are Zariski-dense in $\SL_2$.

On the new bound for the number of solutions of polynomial equations in subgroups and the structure of graphs of Markoff triples

We sharpen the bounds of J. Bourgain, A. Gamburd and P. Sarnak (2016) on the possible number of nodes outside the "giant component" and on the size of individual connected components in the suitably defined functional graph of Markoff triples modulo $p$. This is a step towards the conjecture that there are no such nodes at all.

凸余紧双曲曲面与扩展图的均匀无共振区域

Uniform resonance free regions for convex cocompact hyperbolic surfaces and expanders

本文证明当相关施赖埃尔图构成双侧扩展图时，凸余紧双曲曲面的覆盖族具有均匀谱隙，并给出新的通用无共振区域，推广了 Brooks、Bourgain-Gamburd-Sarnak 等先前结果。